We therefore suggest a mixed explicit implicit approach: only treat the neighborhood of the small cell \(I_k\)  implicit for stability but treat all the cells away from the small cell explicit to keep the cost low. The question is how to change between explicit and implicit time stepping while ensuring conservation and maintaining stability.
May and Berger[1]} examined these questions in the context of cut cell meshes. There one also faces similar meshes as shown in figure REF  that contain tiny cells. The authors suggest to use flux bounding to couple the explicit and the implicit scheme. The work in Ref. [1]} only considered the linear advection equation. Here we extend this to more complicated non-linear flow problems, in particular to two phase flow problems.
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